Rizzo showed that the family of elliptic curves $\mathcal{W}(t) :y^2=x^3+tx^2-(t+3)x+1$, a well-known example of Washington, has root number $W(\mathcal{W}(t))=-1$ for all $t\in\mathbb{Z}$. In this paper we generalize this example and identify the families of small degree on which this phenomenon happens. Motivated by results from David, Bettin and Delaunay (arXiv:1612.03095) and Desjardins (arXiv:1810.12787), we study in detail the two families $\mathcal{F}_s(t):y^2=x^3+3tx^2+3sx+st$ and $\mathcal{L}_{w,s,v}(t): wy^2=x^3+3(t^2+v)x^2+3sx+s(t^2+v)$ and describe necessary and sufficient conditions for which subfamilies of $\mathcal{F}_s(t)$ have constant root number on integer fibres. We further prove similar but partial results on $\mathcal{L}_{w,s,v}(t)$. Our results give examples of subfamilies for which there is rank elevation at integer fibres.
For a non-constant elliptic surface over $\mathbb{P}^1$ defined over $\mathbb{Q}$, it is a result of Silverman that the Mordell--Weil rank of the fibres is at least the rank of the group of sections, up to finitely many fibres. If the elliptic surface is non-isotrivial one expects that this bound is an equality for infinitely many fibres, although no example is known unconditionally. Under the Bunyakovsky conjecture, such an example has been constructed by Neumann and Setzer. In this note we show that the Legendre elliptic surface has the desired property, conditional on the existence of infin…
We consider an elliptic surface $π: \mathcal{E}\rightarrow \mathbb{P}^1$ defined over a number field $k$ and study the problem of comparing the rank of the special fibres over $k$ with that of the generic fibre over $k(\mathbb{P}^1)$. We prove, for a large class of rational elliptic surfaces, the existence of infinitely many fibres with rank at least equal to the generic rank plus two.
We explicitly determine the elliptic K3 surfaces with a maximal singular fibre. If the characteristic of the ground field is different from 2, for each of the two possible maximal fibre types, $I_{19}$ and $I^*_{14}$, the surface is unique. In characteristic 2 the maximal fibre types are $I_{18}$ and $I^*_{13}$, and there exist two (resp. one) one-parameter families of such surfaces.
In this paper, we show that an action on the set of elliptic curves with j= 1728 preserves a certain kind of symmetry on the local root number of Hecke characters attached to such elliptic curves. As a consequence, we give results on the distribution of the root numbers and their average of the aforementioned Hecke characters.
To determine the global root number of an elliptic curve defined over a number field, one needs to understand all the local root numbers. These have been classified except at places above 2, and in this paper we attempt to complete the classification. At places above 2, we express the local root numbers in terms of norm residue symbols (resp. root numbers of explicit 1-dimensional characters) in case when wild inertia acts through a cyclic (resp. quaternionic) quotient.
In a previous article, the author proves that the value of the root number varies in a non-isotrivial family of elliptic curves indexed by one parameter $t$ running through $\mathbb{Q}$. However, a well-known example of Washington has root number $-1$ for every fiber when $t$ runs through $\mathbb{Z}$. Such examples are rare since, as proven in this paper, the root number of the integer fibers varies for a large class of families of elliptic curves. Our results depends on the squarefree conjecture and Chowla's conjecture, and are unconditional in many cases.
By considering a one-parameter family of elliptic curves defined over $\mathbb{Q}$, we might ask ourselves if there is any bias in the distribution (or parity) of the root numbers at each specialization. From the work of Helfgott, we know (at least conjecturally) that the average root number of an elliptic curve defined over $\mathbb{Q}(T)$ is zero as soon as there is a place of multiplicative reduction over $\mathbb{Q}(T)$ other than $-$deg. Recently, Helfgott's work was extended by Desjardins, where she relaxes some of Helfgott's hypotheses and is able to provide unconditional results on the…
Given an elliptic curve over a field of characteristic different from 2,3, its discriminant defines a $μ_{12}$-torsor over the field. In this paper, we give an explicit description of this $μ_{12}$-torsor in terms of the 3-torsion points and of the 4-torsion points on the given elliptic curve. %In addition, we show that such a description involves the Weil pairing in a certain way. As an application, we generalize a result of Coates on the 12-th root of the discriminant of an elliptic curve.
The purpose of the paper is to complete several global and local results concerning parity of ranks of elliptic curves. Primarily, we show that the Shafarevich-Tate conjecture implies the parity conjecture for all elliptic curves over number fields, give a formula for local and global root numbers of elliptic curves and complete the proof of a conjecture of Kramer and Tunnell in characteristic 0. The method is to settle the outstanding local formulae by deforming from local fields to totally real number fields and then using global parity results.
We give an explicit description of the behaviour of the root number in the family given by twists of an elliptic curve $E$ by the rational values of a polynomial $f(T)$. In particular, we give a criterion (on $f$ depending on $E$) for the family to have a constant root number over $\mathbb{Q}$. This completes a work of Rohrlich: we detail the behaviour of the root number when $E$ has bad reduction over $\mathbb{Q}^{ab}$ and we treat the cases $j(E)=0,1728$ which were not considered previously.
Let $C$ be a hyperelliptic curve $y^2 = f(x)$ over a discretely valued field $K$. The $p$-adic distances between the roots of $f(x)$ can be described by a completely combinatorial object known as the cluster picture. We show that the cluster picture of $C$, along with the leading coefficient of $f$ and the action of $\mathrm{Gal}(\bar{K}/K)$ on the roots of $f$, completely determines the combinatorics of the special fibre of the minimal strict normal crossings model of $C$. In particular, we give an explicit description of the special fibre in terms of this data.
We consider the problem of finding $1$-parameter families of elliptic curves whose root number does not average to zero as the parameter varies in $\mathbb{Z}$. We classify all such families when the degree of the coefficients (in the parameter $t$) is less than or equal to $2$ and we compute the rank over $\mathbb{Q}(t)$ of all these families. Also, we compute explicitly the average of the root numbers for some of these families highlighting some special cases. Finally, we prove some results on the possible values average root numbers can take, showing for example that all rational number in …
Let $E/\mathbb{Q}$ be an elliptic curve. For a prime $p$ of good reduction, let $r(E,p)$ be the smallest non-negative integer that gives the $x$-coordinate of a point of maximal order in the group $E(\mathbb{F}_p)$. We prove unconditionally that $r(E,p)> 0.72\log\log p$ for infinitely many $p$, and $r(E,p) > 0.36 \log p$ under the assumption of the Generalized Riemann Hypothesis. This can be viewed as elliptic curve analogues of classical lower bounds on the least primitive root of a prime.
In this article we perform an extensive study of the spaces of automorphic forms for GL(2) of weight two and level N, for N an ideal in the ring of integers of the quartic CM field generated by the twelfth roots of unity. This study is conducted through the computation of the group cohomology of the congruence subgroup of upper triangular matrices modulo N, and the corresponding Hecke action which this space admits. Combining this Hecke data with the Faltings-Serre method for proving equivalence of Galois representations, we are able to provide the first known examples of modular elliptic curv…
We study the elliptic modular surface attached to the commutator subgroup of the modular group. This has an elliptic curve as base and only one singular fibre. We employ an algebraic approach and then consider some arithmetic questions.
Let $E$ be an elliptic surface over the curve $C$, defined over a number field $k$, let $P$ be a section of $E$, and let $\ell$ be a rational prime. For any non-singular fibre $E_t$, we bound the number of points $Q$ on $E_t$ of (algebraic) degree at most $D$ over $k$, such that $\ell^n Q=P_t$, for some $n\geq 1$. The bound obtained depends only on $\ell$, the surface and section in question, $D$, and the degree $[k(t):k]$; that is, it is uniform across all fibres of bounded degree. In special cases, we obtain more specific, in some instances sharp, bounds.
The parity of the analytic rank of an elliptic curve is given by the root number in the functional equation L(E,s). Fixing an elliptic curve over any number field and considering the family of its quadratic twists, it is natural to ask what the average analytic rank in this family is. A lower bound on this number is given by the average root number. In this paper, we investigate the root number in such families and derive an asymptotic formula for the proportion of curves in the family that have even rank. Our results are then used to support a conjecture about the average analytic rank in thi…
We conjecture that, for a fixed prime $p$, rational elliptic curves with higher rank tend to have more points mod $p$. We show that there is an analogous bias for modular forms with respect to root numbers, and conjecture that the order of the rank bias for elliptic curves is greater than that of the root number bias for modular forms.
The parity conjecture has a long and distinguished history. It gives a way of predicting the existence of points of infinite order on elliptic curves without having to construct them, and is responsible for a wide range of unexplained arithmetic phenomena. It is one of the main consequences of the Birch and Swinnerton-Dyer conjecture and lets one calculate the parity of the rank of an elliptic curve using root numbers. In this handbook, we explain how to use local root numbers of elliptic curves to realise some of these phenomena, with an emphasis on explicit calculations. The text is aimed at…
Let E be a one-parameter family of elliptic curves over Q. We prove that the average root number is zero for a large class of families of elliptic curves of fairly general type. Furthermore, we show that any family E with at least one point of multiplicative reduction over Q(t) has average root number 0, provided that two classical arithmetical conjectures hold for two polynomials constructed explicitly in terms of E. The behaviour of the root number in any family E without multiplicative reduction over Q(t) is shown to be rather regular and non-random; we give expressions for the average root…
Given an elliptic surface over a number field, we study the collection of fibres whose Mordell-Weil rank is greater than the generic rank. Under suitable assumptions, we show that this collection is not thin. Our results apply to quadratic twist families and del Pezzo surfaces of degree $1$.
Conjecturally, the parity of the Mordell-Weil rank of an elliptic curve over a number field K is determined by its root number. The root number is a product of local root numbers, so the rank modulo 2 is conjecturally the sum over all places of K of a function of elliptic curves over local fields. This note shows that there can be no analogue for the rank modulo 3, 4 or 5, or for the rank itself. In fact, standard conjectures for elliptic curves imply that there is no analogue modulo n for any n>2, so this is purely a parity phenomenon.
We show the density of rational points on non-isotrivial elliptic surfaces by studying the variation of the root numbers among the fibers of these surfaces, conditionally to two analytic number theory conjectures (the squarefree conjecture and Chowla's conjecture). This is a weaker statement than the one found in a preprint of Helfgott which proves (under the same assumptions) that the average root number is $0$ when the surface admits a place of multiplicative reduction. However, we use a different technique. The conjectures involved impose a restriction on the degree of the irreducible facto…
For an elliptic curve $E$ over a number field $K$, one consequence of the Birch and Swinnerton-Dyer conjecture is the parity conjecture: the global root number matches the parity of the Mordell-Weil rank. Assuming finiteness of $\mathrm{Sha}(E/K)[p^\infty]$ for a prime $p$ this is equivalent to the $p$-parity conjecture: the global root number matches the parity of the $\mathbb{Z}_p$-corank of the $p^\infty$-Selmer group. We complete the proof of the $p$-parity conjecture for elliptic curves that have a $p$-isogeny for $p > 3$ (the cases $p \le 3$ were known). T. and V. Dokchitser have showed …
We show that the Davenport-Stothers inequality from characteristic 0 fails in any characteristic p>3. The proof uses elliptic surfaces over the projective line and inseparable base change. We then present adjusted inequalities. These follow from results of Pesenti-Szpiro. For characteristic 2 and 3, we achieve a similar result in terms of the maximal singular fibres of elliptic surfaces over the projective line. Our ideas are also related to supersingular surfaces (in Shioda's sense).
We consider the following question : given a family over abelian varieties $\mathcal{A}$ over a curve $B$ defined over a number field $k$, how does the rank of the Mordell-Weil group of the fibres $\mathcal{A}_t(k)$ vary? A specialisation theorem of Silverman guarantees that, for almost all $t$ in $C(k)$, the rank of the fibre is at least the generic rank, that is the rank of $\mathcal{A}(k(B))$. When the base curve $B$ is rational, we show, at least in many cases and under some geometric conditions, that there are infinitely many fibres for which the rank is larger than the generic rank. This…
For a global field K and an elliptic curve E_eta over K(T), Silverman's specialization theorem implies that rank(E_eta(K(T))) <= rank(E_t(K)) for all but finitely many t in P^1(K). If this inequality is strict for all but finitely many t, the elliptic curve E_eta is said to have elevated rank. All known examples of elevated rank for K=Q rest on the parity conjecture for elliptic curves over Q, and the examples are all isotrivial. Some additional standard conjectures over Q imply that there does not exist a non-isotrivial elliptic curve over Q(T) with elevated rank. In positive characteristic, …
Given two semistable, non potentially isotrivial elliptic surfaces over a curve $C$ defined over a field of characteristic zero or finitely generated over its prime field, we show that any compatible family of effective isometries of the N{é}ron-Severi lattices of the base changed elliptic surfaces for all finite separable maps $B\to C$ arises from an isomorphism of the elliptic surfaces. Without the effectivity hypothesis, we show that the two elliptic surfaces are isomorphic. We also determine the group of universal automorphisms of a semistable elliptic surface. In particular, this includes…
